Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations

Abbas, Syed; Kavitha, V.; Murugesu, R.

doi:10.1007/s12044-015-0235-6

Cited by 11 publications

(9 citation statements)

References 34 publications

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“…under certain conditions. In [2], S. Abbas, V. Kavitha and R. Murugesu have examined Stepanov-like weighted pseudo almost automorphic solutions to the following fractional order abstract integro-differential equation:…”

Section: Introductionmentioning

confidence: 99%

Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions

Kostić¹

2018

Fractional Differential Calculus

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In this paper, we analyze the existence and uniqueness of generalized weighted pseudo-almost automorphic solutions of abstract Volterra integro-differential inclusions in Banach spaces. The main results are devoted to the study of various types of weighted pseudo-almost periodic (automorphic) properties of convolution products. We illustrate our abstract results with some examples and possible applications.2010 Mathematics Subject Classification. 44A35, 42A75, 47D06, 34G25, 35R11. Key words and phrases. Weighted pseudo-almost periodicity, weighted pseudo-almost automorphy, convolution products, abstract Volterra integro-differential inclusions, multivalued linear operators.The author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia.where A(t) : D ⊆ X → X is a family of densely defined closed linear operators on a common domain D, independent of time t ∈ R, the history u t : (0, ∞] → X defined by u t (·) := u(t + ·) belongs to some abstract phase space B defined axiomatically, and f : R × B → X, g : R × B → X fulfill some conditions. Fractional calculus and fractional differential equations have received much attention recently due to its wide range of applications in various fields of science, such as mathematical physics, engineering, aerodynamics, biology, chemistry, economics etc (see e.g.[5], [24], [32]-[33] and [44]-[45]). In this paper, we essentially use only the Weyl-Liouville fractional derivatives (for more details, we refer the reader to the paper [42] by J. Mu, Y. Zhoa and L. Peng). The Weyl-Liouville fractional derivative D γ t,+ u(t) of order γ ∈ (0, 1) is defined for those continuous functions u : R → X such that t → t −∞ g 1−γ (t − s)u(s) ds, t ∈ R is a well-defined continuously differentiable mapping, by D γ t,+ u(t) := d dt t −∞ g 1−γ (t − s)u(s) ds, t ∈ R;

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Section: Introductionmentioning

confidence: 99%

Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions

Kostić¹

2018

Fractional Differential Calculus

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“…Their discussion is divided into two cases, i.e., : R × → , ( , ) → ( , ) is ( , ])-pseudo almost automorphic and : R × → , and ( , ) → ( , ) is Stepanov-like ( , ])-pseudo almost automorphic. Kavitha et al [62] studied weighted pseudo almost automorphic solutions of the fractional integrodifferential equation…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

“…For this reason, (6) has gotten a considerable attention in recent years (cf. [51][52][53][54][55][56][57][58][59][60][61][62][63][64][68][69][70][71] and the references therein).…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

“…However, in many papers (for instance, [11,[51][52][53][54][55][56][57][58][59][60][61][62][63][64]) on almost periodic type and almost automorphic type solutions to fractional differential equations, to be able to apply the well-known Banach contraction principle, a (locally) Lipschitz condition for the nonlinearity of corresponding fractional differential equations is needed. As can be seen, our results generalize those as well as related research and have more broad applications.…”

Section: International Journal Of Differential Equationsmentioning

confidence: 99%

“…Define a mapping Λ on (R, ) by (62) and prove that Λ has a unique fixed point V( ) ∈ (R, ). Firstly, similar to the proof in Step 1 of Theorem 27, we can prove that Λ is well defined and maps (R, ) into itself; moreover Λ is continuous.…”

Section: International Journal Of Differential Equations 15mentioning

confidence: 99%

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Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Cao

Huang²,

N’Guérékata

2018

International Journal of Differential Equations

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This paper is concerned with the existence of asymptotically almost automorphic mild solutions to a class of abstract semilinear fractional differential equations D ( ) = ( ) + D −1 ( , ( ), ( )), ∈ R, where 1 < < 2, is a linear densely defined operator of sectorial type on a complex Banach space and is a bounded linear operator defined on , is an appropriate function defined on phase space, and the fractional derivative is understood in the Riemann-Liouville sense. Combining the fixed point theorem due to Krasnoselskii and a decomposition technique, we prove the existence of asymptotically almost automorphic mild solutions to such problems. Our results generalize and improve some previous results since the (locally) Lipschitz continuity on the nonlinearity is not required. The results obtained are utilized to study the existence of asymptotically almost automorphic mild solutions to a fractional relaxation-oscillation equation.

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Piecewise Continuous Stepanov-Like Almost Automorphic Functions with Applications to Impulsive Systems

Abbas

Mahto

2019

Current Trends in Mathematical Analysis and Its Interdisciplinary Applications

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Stepanov-like weighted pseudo almost automorphic solutions to fractional order abstract integro-differential equations

Cited by 11 publications

References 34 publications

Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions

Generalized almost automorphic and generalized asymptotically almost automorphic solutions of abstract Volterra integro-differential inclusions

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

Piecewise Continuous Stepanov-Like Almost Automorphic Functions with Applications to Impulsive Systems

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